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User blog:BunsenH/How many days to Celestial? A math "trick"
People have sometimes wondered about how long it takes to complete a Celestials' constellation in the DoF Daily Login game. On average, of course. The answer can take us to a very useful problem-solving method: Dimensional analysis. I learned about it in my last year of high school, I think. In various teaching jobs I've held, I have often found that even many upper-year university students have never learned about it. But it is actually pretty straightforward. It works by a series of multiplications, converting one set of units to another. Let's start with a few things we already know. Each constellation has 24 steps to complete. Each roll moves the Celestial by 11⁄9 steps, on average. And each day, you get one free roll, with an extra roll every 5 days. That "11⁄9 steps per roll" means that on average, 9 rolls give you 10 steps. One way of looking at it is that 10 steps are equivalent to 9 rolls. \rm 10\ steps\ \equiv\ 9\ rolls We can divide both sides of the equation by 10 steps. It is very important to keep the units through all of these calculations! They are an important clue to the steps that must be followed, as well as helping you catch any mistakes. Doing that, we get 1 \equiv And yes, I know that that looks weird. Where it comes in handy is that the weird thing on the right is equivalent to 1. And you can multiply things by 1, as much as you want, without changing them. In particular, we can multiply 24 steps by it. {\rm 24\ steps}\ =\ {\rm 24\ steps} {\rm 24\ steps}\ \equiv\ {\rm 24\ steps}\ \times\ We cancel out the "steps" above and below, and get {\rm 24\ steps}\ \equiv\ {\rm 24}\ \times\ or {\rm 24\ steps}\ \equiv\ {108\over 5} \rm\ rolls Which is the number of rolls which is equivalent (on average) to 24 steps. It's important to note, at this point, that by dividing things the other way, we could get 1 \equiv Which is also perfectly valid and potentially useful for some kinds of calculations. But it isn't useful for this problem, because if we multiply 24 steps by it, the units don't cancel out. In fact, they get quite messy: {\rm 24\ steps}\ \times\ = {240\rm\ steps^2\over\rm 9\ rolls} This is correct, but it doesn't help us. The point I want to get at here is that by keeping the units in the calculation, we can see which way the conversion has to be done. If you just tried to multiply by 9⁄10 -- or is it 10⁄9? -- and slap the units on afterwards, you'd be fairly likely to get the wrong result. Now that we've got the number of rolls, we want to convert that to a number of days. In 5 days, we get 6 free rolls. Or: 1\equiv \rm\ \ \ and\ \ \ 1 \equiv Multiplying a number of rolls by the first of these makes things worse. Multiplying by the second one cancels out the rolls and leaves us with a number of days: {\rm 24\ steps}\ \equiv\ {108\over 5} \rm\ rolls {\rm 24\ steps}\ \equiv\ {108\over 5} \rm\ rolls \times \rm 24\ steps\ \equiv\ 18\ days Which is the answer that we want: 24 steps are equivalent (on average!) to 18 days of free dice rolls. The same kind of calculation can be used in other places in the game. For example, suppose we want to figure out how many coconuts are needed to needed to make a recipe of Cactus Cheesecake. We have: 3 coconuts ≡ 1 coconut milk; 3 coconut milk ≡ 1 coconut cheese; 2 coconut cheese ≡ 1 cheesecake. Or, converting from what we have to what we want: \rm 1\ cheesecak{}e \equiv\ 1\ cheesecak{}e\times {2\ cheese\over 1\ cheesecak{}e} \times {3\ milk\over 1\ cheese} \times {3\ coconuts\over 1\ milk} \rm 1\ cheesecak{}e\ \equiv\ 18\ coconuts All of the multiplications can be written together. All of the units cancel out except the coconuts, which is what we want, and the final answer is 18 coconuts. It is important to be careful about one thing. This only works for conversions that are simple multiplications. An example of a conversion that would fail is something involving temperatures in Fahrenheit and Celsius. Although the sizes of the degrees are a simple multiplication (a change of 5 °C = a change of 9 °F), the two scales are offset. The actual conversion is: Temp in °F = (9⁄5 × Temp in °C) + 32 . If you're dealing with a problem that includes a step like this, you have to break it into pieces and do the odd conversions separately from the rest. It's also rather important that you think about the final number and ask yourself: does that make sense? Is the answer about the size you expect? If not, you should go back and check your work. 18 coconuts to make a cheesecake seems reasonable. 6 coconuts would be much too few; 54 would seem awfully high. But it's also possible that if the answer doesn't match your expectations, it may be your expectations that are wrong. And it's a good thing to correct yourself if they are. This kind of calculation can be useful for more complicated kinds of problems, especially when you're trying to combine bits of information that use different kinds of units. For example, there's Einstein's equation for converting between matter and energy, the famous "E = m''c'' 2". A llittle matter is equivalent to a lot of energy. You would usually be calculating for only a tiny mass, but c is about 300,000 km/s (or 176,000 miles/s). A common unit of energy is the Joule, ‎1 kg⋅m2⋅s-2. If you want to report your results in Joules, you'll need to convert the mass part of your numbers to kg, and your speed-of-light part to m/s. Again, the conversion processes I've shown here will help you to sort these things out without multiplying where you want to divide, or vice versa. Category:Blog posts